Market Power and Cost Efficiencies in Banking1

Pradeep Kumar2

October 1, 2013

Abstract

A merger wave during the last 20 years has led to a decrease in the number of

commercial banks from 12,343 in 1990 to 6,222 in 2012. For anti-trust and regulatory

agencies, evaluating the relative importance of market power and cost efficiencies that

result from a horizontal merger is important. To quantify these two effects, I develop an

empirical model of banking which includes a demand model for differentiated products

that allows market power effects to be calculated and a cost model that quantifies cost

efficiencies from a merger. Since most of these banks operate in more than one market,

incorporating the network structure of branches in the analysis is important. Cost

parameters related to the network structure are estimated using moment inequality

methods. Using the estimated parameters, I simulate mergers between banks of various

sizes. I find that for a merger between 2 small banks (less than 500 branches), cost

efficiencies play an important role in profitability. While for mergers involving a large

bank (more than 500 branches) and a small bank or two large banks, the benefits

accruing from market power are far more than the cost savings if there is substantial

overlap in the networks of the merging banks. Although mergers involving large banks

generate more market power, overall consumer welfare increases as the price effect of

market power gets dominated by the consumer’s preference for a larger bank.

1I gratefully acknowledge the help and support of my dissertation committee: Mark Roberts, Paul Grieco,Jim Tybout and Keith Crocker. I am also thankful for very insightful discussions with Jonathon Eaton, KalaKrishna, Steve Yeaple, Neil Wallace, Robert Marshall, Vikram Kumar, Yao Luo, Felix Tintelnot, HongsongZhang, Van Anh Vuong, Gaurab Aryal, Peter Newberry and other participants of the Empirical IO workshop.All errors are my own. I am also thankful to the Human Capital Foundation for their generous financialsupport.

2Pennsylvania State University, Department of Economics. email:kumarecon7@gmail.com

1 Introduction

Th removal of legal restrictions on intrastate and interstate banking was a gradual process

that culminated with passage of the Riegle-Neal Act in 1994. Since then, the banking

industry has undergone substantial restructuring. This gradual deregulation has led to a

consolidation of banks over the last 30 years that is still ongoing. In 1990, there were

12,343 commercial banks and 2,815 savings FDIC insured banks in the United States.3 In

2012, these numbers had fallen to 6,222 commercial banks and 1,024 savings banks. This

consolidation in the banking industry has been driven primarily by mergers. As the number

of banks has declined, the average size of banks, measured by deposits, has risen from 184

million dollars in 1990 to 978 million dollars in 2012. Since the banking industry is connected

to all other industries and, as we have seen recently, widespread failures can lead to a financial

crisis, it is important to understand the reasons behind the ongoing consolidation and its

impact on the strength of competition and cost efficiencies. This paper will study these

effects of a merger and also quantify its impact on consumer welfare.

Anti-trust and regulatory agencies always screen horizontal mergers for the role of market

power and cost efficiencies as the possible driving forces behind the merger. For a merger

between two conglomerates (firms present in more than one market), market power is often a

geographically local phenomenon while cost efficiencies are realized at the firm level. In this

paper I quantify and compare these two forces. Another aspect that is particular to banking

industry mergers is the consumer’s preference for a large network of branches. Consumers

prefer large network of branches not only due to geographical convenience, but also because

larger banks are deemed safer to bankruptcy risk. As a result, just looking at the price effects

to evaluate a merger could leave out an important effect of merger.4 This paper accounts

for the role of price as well as the preference for large banks in the calculation of consumer

welfare.

I develop a three stage empirical model of competition that captures the long-run effects

of establishing a branch network and short-run effects due to price competition and capital

structure. In the first stage, all banks choose their network of branches. The network decision

comprises the number of branches to open and their location. In the second stage, banks

choose equity capital. Equity capital is needed for three reasons: to satisfy government

regulation constraints, signal safety to uninsured depositors, and acts as an alternative to

deposits for funding loans. In the third stage, banks set deposit interest rates and compete

in a Bertrand competition to collect deposits. Consumers demand deposit services and

3Almost 98% of the banks in the U.S. are FDIC insured.4The 2010 Horizontal Merger Guidelines adopted UPP(Upward Pricing Pressure) as one of the ways to

evaluate a merger which is based on price effects alone.

1

choose banks based upon their characteristics. This choice of timing distinguishes long-run

decisions about network structure from short-run decisions about capital choice and pricing

of deposits.

Structural estimation of the model is done in three stages. First, demand parameters are

estimated using supply-side and demand-side moments jointly. Second, the cost of raising

equity capital is estimated by using moments formed by the first-order condition of equity

capital. Finally, the remaining cost parameters are estimated using the moment inequality

method proposed in Pakes, Porter, Ho and Ishii (2011) , PPHI henceforth. To form inequal-

ities, counterfactual policies are generated using addition and subtraction of branches in

different markets. For inference, I use the PPHI method as well as the generalized moment

selection approach proposed by Andrews and Soares (2011).

The presence of market power in the banking industry is well documented. Prager and

Hannan (1998) find that a reduction in interest rates on local deposit accounts was associated

with horizontal mergers that raised market concentration significantly. Berger, Demsetz, and

Strahan (1999) use data for the 1990s and find a negative relationship between local market

concentration and deposit rates. Simons and Stavins (1998) find that an increase in a local

concentration measure (HHI) leads to a decrease in deposit interest rates.

On the cost side, there is a long debate about the presence of economies of scale in bank-

ing. Studies by Boyd and Graham (1998), Mester (1987), Berger, Hanweck and Humphrey

(1987), and Boyd and Runkle (1993) did not find economies of scale beyond very small banks.

These studies used data from the 1980s and didn’t incorporate risk aspects of banking into

the banking technology. More recently, Hughes, Mester, and Moon (2001), Hughes, Lang,

Mester, and Moon (1996), Hughes and Mester (1998) found significant economies of scale

in most banks when capital structure and endogenous risk taking were explicitly considered

in the analyses of production. My paper also contributes to this stream of literature. None

of these papers controlled for market power at the local geographic level while calculating

economies of scale at the national level. It is possible that the change in profits/costs with

size could be different if we control for local market power. Hence, the conclusions about

scale economies in the existing literature may be misleading as a result of ignoring the market

power effect.

The estimation results show that consumers prefer banks who offer high deposit rates,

have more branches locally and nationwide, and are more capitalized. The cost parameters

estimated through moment inequalities are partially identified and I get set estimates for

these parameters. I allow for the cost function to differ between small (less than 500 branches)

and large banks (more than 500 branches). After controlling for market power, the evidence

for cost efficiencies is weak at best for smaller banks. For larger banks, the cost function is

2

less concave than smaller banks suggesting a decrease in cost efficiencies as banks get larger.

The estimated interest rate to raise equity capital for small banks is 6.03% while for larger

banks the interest rate is 5.24%. This result is reasonable given that large banks are more

diversified and could be perceived as being too big to fail. From an investor’s point of view,

this allows big banks to be considered as a safer investment.

Using the estimated parameters, I simulate mergers between different type of banks to

compare the benefits that accrue from market power versus cost efficiencies. I simulate

mergers in three categories: between two small banks, a small bank and a large bank and

between two large banks. For mergers between two small banks cost efficiencies are found to

play an important role. For a merger between a small and a large bank, the extra revenue

generated by market power is much larger than the cost savings if there is substantial overlap

in the networks of the merging banks. When two large banks merge the market power effect

dominates the cost efficiencies effect. The reason behind these results is the decrease in the

concavity of the cost function for larger banks. The mergers involving larger banks (small-

large bank merger or large-large bank merger) increase consumer surplus if I account for

both price effects as well as consumer’s preference for large networks. Hence, the market

power effect of prices is dominated by a better quality product in mergers involving large

banks. Large network of branches can be assumed to be of higher quality due to two reasons.

First, banking services are easier to access for large networks. Second, larger banks can be

more safer to bankruptcy risk.

The rest of the paper is organized as follows. Section 2 discusses data used for the

analysis. Section 3 introduces the model. In Section 4, I outline the estimation strategy.

Section 5 discusses the estimation results. Section 6 contains counterfactual experiments

and section 7 concludes.

2 Data

2.1 Data Sources

The data used in this paper is a cross-section of commercial banks from 2006.

Data is taken from three sources. Information on bank ownerships, location of branches

and deposits is taken from Summary of Deposits at the Federal Deposit Insurance Corpo-

ration (FDIC). The variables in the FDIC data can be divided into three categories: Bank

Holding Company (BHC) variables, institution (bank level) variables, and branch variables.

Bank holding companies (BHC) are at the top of hierarchy, with banks in the middle level

and branches are at the bottom. A bank holding company (BHC) is a company that con-

3

trols one or more banks. There are also banks that are not owned by any BHC. All BHCs

in the U.S. are required to register with the Board of Governors of the Federal Reserve Sys-

tem whereas non-BHC banks can function under the supervision of the Comptroller of the

Currency or the Federal Deposit Insurance Corporation. A BHC needs a separate charter

for each bank.5 For example, Bank of America may have many bank charters and multiple

branches across the country but it will have only one BHC. In this paper, the decision maker

is a BHC. Some small banks are not registered as a BHC and for them the bank is the

decision maker. For the rest of the paper, banks and bank holding companies (BHC) are

used interchangeably.

I also use bank level data taken from the Call Reports available at Federal Reserve Bank of

Chicago. Call reports contain information on interest expenses on deposits, interest revenues

from loans, equity capital, total employees and wages. Using the hierarchy information in

the FDIC data, the bank level data from the Call Reports is aggregated into BHC level data.

The interest rate on deposits is calculated as a ratio of interest expenses to total deposits.

Similarly, the interest rate on loans is calculated as a ratio of interest revenue to total loans.

I use wages and employees per branch as instruments in the demand estimation.

Data on demographic information such as population and income is taken from the

Bureau of Economic Analysis (BEA). Total income in a market is used to calculate the

outside good in the demand estimation where consumers choose banks for deposit services.

I also use market level income as a measure of market size to construct the branch density

variable used in demand estimation. Market level population is used to form weighting

function used in the moment inequality estimation.

2.2 Market and Data Summary

A market is defined as a Metropolitan Statistical Area (MSA). Antitrust analysis has

relied on the definition of a banking market at the MSA level. Using data from the Survey

of Consumer Finances, Amel and Starr-McCluer (2001) find that households obtain 90% of

the checking accounts, savings accounts and certificates of deposits within the local market.

Kwast, Starr-McCluer and Wolken (1997) find that over 94% of small businesses use a local

depository institution.

The sample covers 353 MSAs and 4,316 firms.6 Out of the 4,316 decision makers, 3,194

5Becoming a bank holding company makes it easier for the firm to raise capital than as a traditionalbank. The holding company can assume debt of shareholders on a tax free basis, borrow money, acquireother banks and non-bank entities more easily, and issue stock with greater regulatory ease. The downsideincludes responding to additional regulatory authorities.

6In total there are 366 MSAs. Small MSAs that do not have any commercial banks are not considered inthe analysis.

4

are BHCs and 1,122 are banks that are not part of a BHC. The average number of BHCs in

a market is approximately 23. The smallest number of BHCs in a market (El Centro, CA)

is 5, while the market with the most BHCs, Chicago-Joliet-Naperville, IL-IN-WI, has 227.

As banks get larger in size they set lower deposit interest rates. The correlation coefficient

between the log of deposit interest rate and the log of the number of branches is -0.0493

(0.0012).7 This provides some preliminary evidence that bigger banks may exercise market

power by setting lower deposit interest rates.

There are 8,252 market-bank observations in the data. The following table provides a

size distribution,

Size(# branches) Total Banks Average Assets(million $’s) per bank1 1, 202 129

2− 10 2, 519 34311− 100 527 2, 165101− 500 51 43, 600

500+ 17 295, 000Total 4, 316 2, 175

Table 1: Size Distribution

The size distribution is skewed towards smaller banks, where size is defined by the number

of branches and there are very few large banks. This suggests that larger banks may have a

different incentive structure than smaller banks. Cohen and Mazzeo (2007) treat single and

multi-market banks separately to assess competition among retail depository institutions in

rural markets. Ramiro (2009) finds that there are significant revenue and cost differences

between single and multi-market banks. Motivated by this, I distinguish between small and

large banks in the construction of the cost function. The cost function in this paper depends

on the total size, measured as the number of branches in the network, and concavity of this

function gives us a measure of the cost efficiencies.

The cut-off of small versus large banks is chosen at 500 branches (log(500 + 1) ≈ 6.23)

using the scatter-plot in figure 1. There are 17 banks above this cut-off.8 It is also im-

portant to separate the large banks from a policy standpoint. One of the objectives of the

recent Dodd-Frank act was to immunize the economy from the failing of such large banks.

Sometimes these large banks are also referred as too-big-to-fail banks.9

7P-values are reported inside the parenthesis.8Although the cut-off between small and large banks is ad-hoc, there is no reason to believe that this

choice can alter the qualitative results of this paper.9The stated aim of the Dodd-Frank legislation is: To promote the financial stability of the United States

by improving accountability and transparency in the financial system, to end “too big to fail”, to protect

5

Figure 1: Scatterplot between log(Assets) and log(branches). Each data point represents afirm (bank or BHC).

Overall there are two patterns worth noting in this section. First, the data provides some

preliminary evidence of market power. I take that into account by developing a model where

banks competitively collect deposits. Second, there maybe a need to distinguish between

small and large banks in their technology. I address this issue by distinguishing between the

small and large banks in the cost function. I incorporate these insights into the model.

3 Empirical Model of Consumer Demand and Firm

Choice

I develop a model of consumer behavior and firm choice. Consumers choose banks for their

deposits to maximize utility. Their utility depends upon the returns on deposits, security

from bank failure and convenience of availing banking services. Banks choose the network of

branches, equity capital and deposit interest rate in a three-stage game to maximize profits.

The model will be used to quantify market power, cost efficiencies and consumer welfare in

the merger simulations. The model has local market competition as the source of market

power while the cost efficiencies are at the firm-level.

the American taxpayer by ending bailouts, to protect consumers from abusive financial services practices,and for other purposes.

6

3.1 Consumer Demand

Consumers either save money in a bank or spend it on the outside good. Consumers

save money by choosing a bank for deposit services based upon the bank’s characteristics.

Consumer i’s utility from deposit services of bank j in market m is:

Uijm = θ1Pdjm + θ2djm + θ3ONEj + θ4(

kjnj

) + θ5log(nj + 1) + θ6 + ξjm + εijm,

where P djm is the deposit interest rate and djm is the branch density of bank j in market

m. Branch density (djm) is defined as the ratio of the number of branches in a market m of

bank j to total income (a measure of market size) in the market m.10 The consumers have a

preference for branch density because they incur a disutility from distance traveled for their

deposit services, which was most recently shown by Ho and Ishii (2010). ONEj is the dummy

variable for one-branch banks. Business model of one branch banks and multi-branch banks

could be very different with the latter’s main focus on the expansion of their branch network.

Therefore, I allow for them to have different effects on the consumer’s utility. (kjnj

) is the

ratio of equity capital (kj) to the total number of bank branches (nj). The capital-asset

ratio which banks use as a signal for safety to uninsured depositors (Hughes and Mester

(1998)) is proxied by (kjnj

).11 I allow for the bank’s size (nj) to enter the consumer utility

function logarithmically since the size distribution is highly skewed. This term captures the

importance of branches outside the depositors’ market as I already conditioned the utility

on branch density inside the market. Consumers care about a bank’s size for two reasons.

First, larger banks are perceived safer as compared to smaller banks with respect to the

bankruptcy risk. This perception can develop due to more diversified portfolio of a large

bank or by observing government bailouts of larger banks in the past. Second, consumers

who travel a lot will care more about the total number of branches of a bank.12 When

simulating a merger, this preference for total branches owned by a bank will give positive

utility to the consumer. Unobserved bank-market quality is denoted by ξjm which includes

characteristics like the number of service counters in a branch, quality of employees and time

taken to serve a customer. Also, ξjm will measure the firm-market fixed effects that are not

captured by the observed variables. Measurement error is denoted by εijm which is assumed

to be type-1 extreme value distributed error.

10Using population as a measure of market size provides similar results.11If I use capital-asset ratio instead of

kj

nj, it would make the market share equation endogenous to solving

an implicit integral equation, as in equilibrium: assets=deposits+capital. This makes the level of demandof deposits a function of deposits itself in the integral equation.

12Since I don’t have data on the number of ATM machines, total number of branches acts as a proxy forit. I am making the assumption that number of branches and number of ATMs are positively correlated.

7

Market share of a bank is calculated as the total deposits of a bank in a market divided

by the total income in that market.13 Suppose there are j = 1, 2, ..., J banks in a market m

and let the outside good be denoted by 0. Then the mean utility of a bank j in market m,

δjm, can be defined as

δjm = θ1Pdjm + θ2djm + θ3ONEj + θ4(

kjnj

) + θ5log(nj + 1) + θ6 + ξjm.

Using the logit error assumption on εijm, I can define the market share, sjm, as

sjm =exp(δjm)

1 +∑J

k=1 exp(δkm),

where the utility of the outside good is normalized to zero.

3.2 Supply side

Banks generate revenues from loans and incur costs on interest expenses on deposits,

equity capital, labor, and physical capital expenditures. A bank’s maximization problem is

maxnj ,kj ,P d

jm

Πj

s.t. Πj = LjPlj −

∑m

Dmsjm(θ)P djm − C(nj, kj)

Lj ≤∑m

Dmsjm(θ) + kj

G(kj, Lj) ≥ ∆,

where Lj are the total assets of a bank, P lj is the average interest rate on assets, Dm

denotes total deposits in market m and sjm(θ) is the market share of bank j in market m. kj

is the equity capital of a bank i.e. bank’s own money at stake. The interest rate on assets,

P lj , is allowed to be correlated with ξjm.14 This is important as a bank with high quality

(or high brand effect) is likely to offer loans with higher interest rates. The cost, C(nj, kj),

consists of two components: labor/physical capital cost and the cost of equity capital. The

labor and physical cost incurred by a bank is for loan services, deposit services, advertising

expenses and risk management. I assume this cost to be a function of the number of branches

(nj). Note that this cost is independent of the location of these branches. It is intuitive to

think that a branch in a bigger market should have higher costs compared to one in a smaller

13An alternative would be to define market shares in terms of bank accounts. The data on accounts is notavailable by market but is at the firm-level. Dick (2002) defines market shares in terms of accounts ratherthan deposits by allocating the total accounts of a bank to each market. She found that results are robustto this alternative definition.

14In other words, for demand estimation, P lj is not used as an instrument.

8

market. Although this is true to some extent, the objective of this paper is to find scale

efficiencies which are usually assumed at the firm level. These efficiencies are realized by a

reduction in expenses at the firm level such as advertising expenditures or risk management

expenses which are common across many markets.

The first constraint captures feasibility i.e. total loans made by a bank can be funded

either by deposits or by equity capital. The second constraint is a regulation constraint. As

per the guidelines of the Board of Governers of Federal Reserve Bank, all chartered banks

in the U.S. should have a capital-asset ratio above a threshold.15 In practice, all banks are

well above the limit, hence this constraint never binds in the data.

Apart from the regulation constraint, there are two other roles for equity capital. First, it

is a source of funding for loans as an alternative to deposits. Hence, labor cost and physical

capital cost spent to collect deposits are affected by the level of equity capital. So failing to

condition on equity capital can bias the cost parameters. Second, banks use equity capital as

a signal of safety to uninsured depositors.16 This is accounted for in the demand model. The

role of equity capital in the cost function of a bank was first noted by Hughes and Mester

(1998).

I assume a parametric form for the bank cost function,

C(nj, kj) = β1nj+β2n2j+β3I(nj > X)(nj−X)2+[βS4 I(nj ≤ X)+βL4 I(nj > X)]kj+γj+νj,nj

,

where γj is the bank level cost shock unobserved to the researcher but observed by the

bank when it makes it decisions. The measurement error or specification error is denoted by

νj,nj.17 The parameter β3 applies for large banks (nj > X) only, where X is the chosen size

cut-off between small and large banks. For small banks, the sign of β2 will determine the

presence or absence of cost efficiencies. For large banks, both β2 and β3 measure concavity

of the cost function. The parameters βS4 and βL4 measure the interest rate paid on equity

capital by small and large banks, respectively. As discussed in Section 2, small banks and

large banks are distinguished in the cost function. The difference in production technologies

between small and large banks will be captured by the parameter β3. Also, any difference in

the cost of external funding will be measured by βS4 and βL4 . Note that although the bank

15As per the guidelines of Board of Governors of Federal Reserve Bank, all chartered banks in US have tosatisfy three constraints: (1) Tier 1 capital / Risk-adjusted assets > 6% (2) Total capital / Risk-adjustedassets > 10% (3) Tier 1 capital / Average total consolidated assets > 5% . More than 99 % of the banks arewell above the threshold limits.

16Data shows a negative relationship between size and capital-asset ratio. Hence, there is some evidencethat larger banks need to signal less about their safety than smaller banks.

17This error term is similar to the non-structural error term in Pakes, Porter, Ho and Ishii (2011)

9

will choose a network of branches, the cost function only depends on the total number of

branches. The network effects are accounted for through the revenue side.

Using the demand and supply side of the model discussed above, I simulate and measure

the market power and cost efficiency trade-off that will be relevant to evaluating a bank

merger. Since market power is a geographically local phenomenon, it will be a function of

the network of branches of the two merging banks. Market power will be measured as the

difference in profits between the merged entity at new prices (deposit interest rate) and the

two merging banks at pre-merger prices. The profit of the merged entity will include the

joint effect of market power and consumer preferences for large networks. To understand

the different forces, I need to isolate the effects to study them separately. Cost efficiencies

are measured as the difference in cost expenditure between the merged entity and the two

merging banks. Since cost efficiencies are realized at the firm-level the network structure is

irrelevant here, only the total size of the network matters for the cost efficiency calculation.

I also quantify the change in consumer welfare generated by a merger.

In equilibrium, the profit function of a bank can be simplified by substituting the feasi-

bility constraint at equality (assets=liabilities) and using the parametric cost function,

Πj =∑

mDmsjm(θ)(P lj − P d

jm) + P ljkj − β1nj − β2n2

j − β3I(nj > X)(nj −X)2 − [βS4 I(nj ≤X) + βL4 I(nj > X)]kj + γj + νj,nj

.

3.3 Timing of the Firm Choices

There are the three stages in the game between banks. In the first stage, banks choose

the network of branches (nj) i.e. how many branches to open and where to locate them.

This decision is made simultaneously by all banks. In the second stage, banks choose their

equity capital (kj). Equity capital is chosen at the firm level. In the third stage, banks

compete for deposits in a Bertrand competition within each geographic market and choose

interest rates (P djm).

4 Estimation

The model is solved and estimated using backward induction. In the first stage, demand

parameters are estimated. The second stage involves estimation of the cost parameter (βS4

and βL4 ) on equity capital. In the third stage, cost parameters related to the network structure

are obtained using moment inequalities.

10

4.1 First Stage: Demand Estimation

Demand parameters are estimated using demand-side and supply-side moments jointly.

In the final stage of the game, bank’s compete for deposits and set their deposit interest

rates (prices). I assume that the competition for deposits in each market is independent of

the competition in other markets. Under this assumption, the first order condition w.r.t.

P djm is

Dm∂sjm(θ)

∂P djm

(P lj − P d

jm)−Dmsjm(θ) = 0 ∀m.

The first order condition above represents a trade-off: an increase in deposit interest rate

will increase expenditure on all the existing deposits but increased deposit interest rate will

also increase the market share of a bank for deposits and hence increase the revenue from

loans. Using the fact that a consumer’s utility has a logit error term (εijm), I can write the

slope of the market share function w.r.t. price as∂sjm(θ)

∂P djm

= θ1sjm(1− sjm). Substituting this

into the above first order condition I get,

Mjm(θ) ≡ θ1(1− sjm)(P lj − P d

jm)− 1 = 0 ∀m (1)

This equation acts as the basis for the supply side moment. I denote this equation by Mjm(θ).

To derive demand side moments, I need to calculate ξjm(θ). Since a consumer’s utility

has a logit error, the demand shock can be solved for explicitly as

ξjm(θ) = ln(sjm)− ln(s0m)

where s0m is the market share of the outside option.

Demand side moments are derived by finding instruments (Zjm) which are un-correlated

with bank-market shocks (ξjm) to consumer utility.18 Specifically, I need instruments for

prices (P djm) and equity capital (kj). The correlation between prices and unobserved quality

(ξjm) is obvious. In the second stage, banks choose kj as a function of ξjm and other variables,

hence they may be correlated. Intuitively, a bank with higher quality, may choose lower kj

as both are substitutes in the consumer’s utility. Instrumental variables used for prices are

cost shifters (wages and employees per branch) and rival firm characteristics (number of rival

banks in a market, number of rival branches in a market, size of rival banks). Instrumental

variable used for kj is wages since higher wages correspond to a higher cost of collecting

18This moment is developed in the spirit of Berry(1994), where the author finds that matching marketshares to estimate demand parameters directly can lead to an endogeneity bias because price can be correlatedto the unobserved quality term.

11

deposits, which would lead firms to choose a higher level of equity capital. Overall, Zjm

includes wages, employees per branch, rival firm characterstics, djm, ONEj and log(nj + 1).

Hence, the estimating GMM equation is given by,

E

[Mjm(θ)

Zjmξjm(θ)

]= 0

There are 11 moment conditions I use to estimate 6 parameters. Hence, the system

is over-identified. I use two step GMM with an optimal weighting matrix to estimate the

parameters.

4.2 Second Stage

In the second stage, banks choose equity capital (kj). Assuming risk-neutrality, a bank’s

equity capital choice is modeled as

maxkj

∑m

Dmsjm(θ)(P lj − P d

jm) + P ljkj − C(nj, kj)

s.t. G(kj, Lj) ≥ ∆

θ1(1− sjm)(P lj − P d

jm)− 1 = 0.

The last constraint is the FOC of Bertrand competition and appears as banks take into

account the effect of equity capital choice on the deposit rates chosen in the last stage. The

first constraint is the regulation constraint and almost all banks in the data easily satisfy it.

Hence, I can assume an interior solution to the above maximization problem. On substitut-

ing the FOC of P djm into profits yields,

Πj =∑

mDmsjm(θ)

θ1(1− sjm(θ))+ P l

jkj − C(nj, kj).

I can write the FOC of kj i.e.∂Πj

∂kj= 0 as,

∑m

Dm

θ1[

∂sjm∂kj

1− sjm+

sjm∂sjm∂kj

(1− sjm)2] + P l

j −∂C(nj, kj)

∂kj= 0.

On simplifying the FOC of kj I get19

∑m

Dm

θ1[

θ4sjmnj(1− sjm)

] + P lj = βx4 , (2)

19The details of the calculation can be found in the appendix 8.4.

12

where x={S,L} corresponds to small or large banks. The first term on the left corresponds to

the marginal revenue from uninsured depositors and the second term on the left is the revenue

per dollar of loans generated from loans that were funded by equity capital. Parameters βS4

and βL4 measure the marginal cost of equity capital to small and large banks respectively.

Assuming a mean zero measurement error in the above equation, parameters βS4 and βL4 can

be estimated by taking expectation of both sides of the equation

βx4 = E[∑

mDm

θ1[

θ4sjmnj(1−sjm)

] + P lj ]

In a more general setup, the marginal cost of equity capital could be measured as a

function of observed variables such as size and geographic diversification. Geographic di-

versification measures the risk in the loan portfolio of bank as well as the risk in collecting

deposits (Aguirregabiria, Clark and Wang (2011)), hence there is an incentive for investors to

look at this aspect. The number of branches a bank has is an important decision criteria for

investors as larger banks can be perceived as too big to fail. To understand the relationship

between the cost of equity capital, the bank’s size and its diversification, I estimate these

relationships using OLS regression. I measure geographic diversification by the number of

markets a bank is present in.

4.3 Third Stage

The cost function parameters β1, β2 and β3 are estimated in this stage. The estimates

quantify how the network of branches affect the bank’s fixed cost and will enter the calcula-

tion of the cost efficiencies. To this end I employ the moment inequality estimator proposed

by Pakes, Porter, Ho and Ishii (2011) (henceforth PPHI).

Let Ij be the information set of the bank when it chooses the network of branches (nj).

The information set Ij consists of {ξjm}j=J,m=Mj=1,m=1 , P l

j and γj.20 The first component of Ij,

{ξjm}j=J,m=Mj=1,m=1 , implies that banks know their fixed brand effect in each market and that of

other banks when making the network choice. This means that a bank deciding to open

a branch in a market knows how it will be perceived by the consumers conditional on the

other observed variables in their utility function. The second component of Ij, Plj , means

that banks know what kind of asset portfolio they will be investing in before deciding where

to open branches and how many to open. For example, a lower P lj means a bank is targeting

a low risk-low return portfolio and knowing this it will open branches in markets with more

predictable, low-return industrial activity. Another way to rationalize this assumption is

to assume that banks are targeting a fixed level of returns on their assets before making

20The PPHI estimator allows for the possibility of the information set to remain unspecified. So, inprinciple there could be variables which are unobserved to the econometrician but are in Ij .

13

any entry decisions. The third component of Ij, is the firm level cost shock, γj, and it

corresponds to the management practices and organization structure that are specific to a

particular bank. This implies that a bank making an entry decision knows about its specific

management practices.21

A bank maximizes its expected profits,

maxnj

E[Πj|Ij]

where Πj is the firm-level profits previously specified in section 3.2. A choice of network

consists of the number of branches to open and their locations across markets. A bank can

choose to have 0, 1 or more branches in any market. The expectation arises because of the

uncertainty in the bank’s observed profit at the time decisions are made. This uncertainty

arises due the randomness in the decision of the rival banks, n−j. The randomness could be

due to the error term, νj,nj, or due to the presence of mixed strategies.

The profit function of a bank j is

Πj =∑m

Dmsjm(θ)(P lj − P d

jm) + P ljkj − β1nj − β2n2

j − β3I(nj > X)(nj −X)2

−[βS4 I(nj ≤ X) + βL4 I(nj > X)]kj + γj + νj,nj(3)

The location of branches affects the profits through the first term. For example, consider

two banks with the same number of branches and similar in all aspects except for the location

of these branches. These two banks collect different deposits in each market because the

market share, sjm, depends on the branch density (djm) in that particular market alongside

other variables. This will lead to these two banks having different profits.

Note that the network choice parameters cannot be estimated directly using the maximum

likelihood estimation. This is due to the fact that the possible network choices a bank has

in the maximization problem are way too large compared to the number of choices observed

in the data.22 Hence I employ the moment inequality method to partially identify the

parameters.

A necessary condition for any Nash equilibria is that the expected profits from choices

observed in the data are greater than any other feasible alternate choice. This forms the basis

of the moment inequality estimation method. I construct an expression for the difference in

21If {ξjm}j=J,m=Mj=1,m=1 was not in the information set, to form moment inequalities I would have to simulate

the expected levels of this variable from a distribution approximated from the demand estimation.22I observe 4,316 firm choices in data. The possible network choices for 4,316 firms in 353 markets with

maximum allowed branches in a market as 500 are: 4316× 353500. Hence it is almost impossible to directlyestimate the parameters with such sparse information.

14

profits for the two policies and then take moments of this differenced profit expression to

form the estimating inequalities. Using the profit function for the choice observed in the data

(nj) and some other alternate policy (n′j), so that (nj − n

′j = t), I can difference the profits.

The alternate policy, n′j, involves addition or subtraction of a fixed number of branches, t,

from the existing network. Following is the differenced profit equation,

∆Πj = ∆Yj(nj, n′

j, n−j)− β1(nj − n′

j)− β2(n2j − n

′2j )

−β3[I(nj > X)(nj −X)2 − I(n′

j > X)(n′

j −X)2] + νj,nj ,n′j, (4)

where νj,nj ,n′j

= νj,nj− νj,n′

jand ∆Yj(nj, n

′j, n−j) is the part of the differenced profit

function which doesn’t contain any parameters to be estimated (because they have already

been estimated in the first and second stage) :

∆Yj(nj, n′

j, n−j) = [∑m

Dmsjm(θ)(P lj − P d

jm) + P ljkj − [βS4 I(nj ≤ X) + βL4 I(nj > X)]kj]

−[∑m

Dms′

jm(θ)(P lj − P

′djm) + P l

jk′

j − [βS4 I(n′

j ≤ X) + βL4 I(n′

j > X)]k′

j].

To evaluate (∆Yj) all the endogenous variables need to be evaluated under the alternate

policies: market shares (s′jm), deposit interest rates (p

′djm) and equity capital (k

′j). Market

share and deposit rates are solved as a system of equations using the first order conditions

for deposit rates (see equation (1)) with the alternative network structure. Equity capital

is estimated non-parameterically from the first order condition for equity capital choice (see

equation (2)) using a third degree polynomial function. In the above equation, the error

term, νj,nj ,n′j, is attributed to measurement error or specification error.

We can simplify the differenced profit function as,

∆Πj = ∆Rj(nj, n′j, n−j) + νj,nj ,n

′j,

where ∆Rj is defined as,

∆Rj(nj, n′

j, n−j) = ∆Yj(nj, n′

j, n−j)− β1(nj − n′

j)− β2(n2j − n

′2j )

−β3[I(nj > X)(nj −X)2 − I(n′

j > X)(n′

j −X)2].

Using the above notation a moment function can be formulated as,

S(β) = E[h(n′j;nj, Ij)∆Rj(nj, n

′j, n−j)]

where h(n′j;nj, Ij) is the weighting function defined below using instruments zj,

15

h(n′j;nj, Ij) =

{g(zj) if nj − n

′j = t

0 otherwise

where zj ∈ Ij are demand shifters which are independent of cost shocks. Refer to the

appendix 8.1 for the satisfaction of the sufficiency conditions for the PPHI estimator.

Essentially, I am looking for parameters that satisfy S(β) ≥ 0. The dimension of the

moment function is dim(h) × dim(∆Rj) i.e. I have more moment restrictions if I have more

weighting functions or have more alternate policies. Since banks are interacting agents in

a particular market, I make use of this by forming the sample analog of the moment by

averaging over banks in a market, followed by averaging over all markets:23

s(β) =1

M

∑m

1

Jm

∑jm

h(n′

j;nj, Ij)∆Rj(nj, n′

j, n−j). (5)

The following equation forms the basis for estimation

β̂ = {β : β ∈ arg minβ||(s(β))−||}, (6)

where (·)− = min(·, 0) and β̂ is the set of identified parameters. The norm used is L1.24 A

usual concern with the set identification approach is that the identified set may potentially

be so large that it is uninformative. In practice, this is taken care of by imposing a large

number of moment restrictions. This is the case here.

In the objective function, there are two categories of moment conditions. The first set

of moment conditions only apply to the small banks. The second set of moment conditions

apply to all banks. This choice of moments is crucial for restricting the set size of β3. The

first set of moments provides identifying power only for β1 and β2. With β1 and β2 restricted

by the first set of moments, the second set of moments identifies β3. Also, this choice of

moments is important as small banks may behave differently from the large banks.25

I use three methods to do inference: PPHI inner and outer confidence interval meth-

ods and point-wise generalized moment selection method proposed by Andrews and Soares

(2011). In the literature for inference of partially identified models, a distinction is made

between constructing a confidence interval for the identified set versus a confidence interval

for the true parameter. The first two methods from PPHI fall in the first category in which

23This idea was previously used by Ishii(2007).24Results with L2 norm are almost the same.25Note that this choice of moments could not be done another way i.e. one set of moments targeting large

banks and second set of moments targeting all bank or small banks. This is because there are only 17 largebanks and I wont have enough observations to average over.

16

confidence intervals are for the extreme points of the identified set. Inference using gener-

alized moment selections lies in the second category where I can do inference point-wise.

Using generalized moment selection method, I can do inference for a point outside of the

estimated set also. The algorithm used for the generalized moment selection inference can

be found in appendix 8.2.

A growing industrial organization literature considers partially identified models using

moment inequalities for estimation. Holmes (2011) studies diffusion of Walmart using the

PPHI estimator. Ishii (2008) studies the effect of ATM surcharges on competition and welfare

using moment inequalities also using the PPHI estimator. Ciliberto and Tamer (2009) study

an entry game among airlines using a partially identified model. Ellickson, Houghton and

Timmins (2012) use moment inequality method to estimate chain economies in the retail

industry.

4.4 Weighting functions and alternate policies

To decrease the size of the identified set, I use several moment restrictions. Hence, to

increase the number of moment conditions in the objective function (equation (6)), I can

either increase the number of weighting functions (h(·)) or increase the number of alternate

policies. For each combination of a weighting function and an alternate policy, I can form a

moment inequality. The weighing functions used for estimation are:

1. Constant function

2. I[ Mean population of the markets bank j is present in ≥ population of market m]

3. I[ Mean population of the markets bank j is present in ≤ population of market m]

I use two weighting functions other than the constant function. The second weighting

function in the above list, includes markets which are larger (in terms of population) than

the average market the bank is present in while forming the moments. Using this weighting

function includes more of the larger markets in the moment conditions. The third weighting

function is just the opposite and includes markets which are smaller than the average market

for a particular bank.

The alternate choice of network used to form moments deviates from the choice in the

data only marginally so that the estimated bounds can be tighter. When the alternative

policy used (n′j) only differs marginally from the actual policy (nj), we are closer to the

trade-off the banks may have faced when making the entry decision. For example, a bank

with 100 branches in the data is more likely to have considered the possibility of opening

17

95 or 105 branches, rather than 50 or 200 branches. Also, since profits are calculated by

summing over markets, I don’t add up the estimation error in demand parameters in the

calculation of ∆R(·) in equation (5) if I change the policy in the data only marginally.

Alternate choice of the network of branches has to involve either adding new branches or

removing existing branches. Otherwise, if I just change the location of existing branches in

the data to form an alternate policy without addition/subtraction of branches, the terms

involving parameters to be estimated will vanish (see equation (4)). Policies that involve

adding branches help to bound the marginal cost from below. Similarly, policies that involve

subtracting branches bound the marginal cost from above. Hence, using these two kind

of alternate policies jointly gives us tighter bounds for the cost parameters. In principle,

any alternative policy will satisfy the inequalities since they correspond to the necessary

condition of the Nash equilibrium.

I have to decide which markets to add/subtract branches, for construction of the alternate

policies. For the moments involving all banks, I add/subtract branches only in the markets

where the banks have the largest presence. In the data, large urban areas see more growth

in branching while in the smaller cities the number of branches has been almost stagnant.26

This fact suggests that banks are more interested in their choices regarding large markets.

So, I create alternate policies that affect larger markets which usually are the markets where

a bank has a large presence. For the moments involving small banks only, I add/subtract

branches in markets where the banks have largest presence as well as in markets where

the banks have least presence. This captures the fact that some of the small banks only

target a small region and are in the growing phase. This is reasonable given the skewed size

distribution in the data.

The alternate policies used in the moment conditions that involve all banks are,

1. Adding 1 branch in a market where the bank has its largest presence.

2. Subtracting 1 branch in a market where the bank has its largest presence.

3. Adding 1 branch each in the 5 markets where the bank has its largest presence.

4. Subtracting 1 branch each in the 5 markets where the bank has its largest presence.

5. Adding 1 branch each in the 10 markets where the bank has its largest presence.

6. Subtracting 1 branch each in the 10 markets where the bank has its largest presence.

The alternate policies used in the moment conditions that involve small banks only are,

26Refer to FDIC 2006 FYI bulletin for details.

18

1. Adding 1 branch in a market where the bank has its largest presence.

2. Subtracting 1 branch in a market where the bank has its largest presence.

3. Adding 1 branch each in the 2 markets where the bank has its largest presence.

4. Subtracting 1 branch each in the 2 markets where the bank has its largest presence.

5. Adding 1 branch each in the 2 markets where the bank has its least presence.

6. Subtracting 1 branch each in the 2 markets where the bank has its least presence.

7. Adding 1 branch each in the 10 markets where the bank has its largest presence.

8. Subtracting 1 branch each in the 10 markets where the bank has its largest presence.

Some small banks go out of business with alternate policies involving subtraction of

branches. Profits of these banks are equated to zero under alternate policies.

5 Results

This section is divided in two parts. The first part contains results from the demand

estimation. These results show the presence of market power and consumer’s preference

for a large network of branches. The second part contains results from estimation of the

cost function. The parameter on equity capital will measure the cost of external funding.

Remaining parameters in the cost function will measure the magnitude of cost efficiencies.

5.1 Demand Parameters

Demand parameters are estimated from 353 markets using demand and supply side mo-

ments jointly. The estimated parameters are reported in table 2. The parameter on the

deposit rate, θ1, is 24.1617 suggesting that consumers prefer high deposit rates.27 The pa-

rameter on the branch density variable, θ2, is 22.2884 implying a preference for the distance

traveled by depositors. It strengthens the finding of Ho and Ishii (2010) that consumers

incur a disutility from distance traveled for their deposit services. The value of -0.5817 on

θ3 indicates consumers aversion towards 1-branch banks for their deposit services. This sup-

ports the fact that the 1-branch banks are not after deposits and their business model may

be different. This is inline with the finding in Dunne, Kumar and Roberts (2012), where

27All the variables in demand estimation are scaled to be of the same order for numerical stability. Hencethe magnitudes of the parameters doesn’t have any direct meaning.

19

the authors find that 1-branch bank’s main source of revenue is the non-interest income.

The parameter on the capital-size ratio, θ4, is estimated to be 2.3054 suggesting that con-

sumers prefer banks which are more capitalized. One possible reason for this parameter to

be just significant could be that part of the population (insured depositors) don’t care too

much about the safety of their deposits. The parameter on size, θ5, is 0.1561 and significant

suggesting that consumers favor banks with more branches, although their branches may be

outside of the market. Depositors have an incentive to look for signals about the safety of

their deposits and we account for it by including size of the bank and the capital-size ratio

which acts as a proxy for capital-asset ratio. Standard errors reported are heteroscedasticity

robust standard errors.

Demand and supply jointly Demand and supply jointly(with IV) (without IV)

Variables Parameter Value Std. Error Value Std. ErrorPrice θ1 24.1617 0.1220 24.1317 0.1282

Branch Density θ2 22.2884 0.5789 22.26371 0.58711-Branch Dummy θ3 -0.5817 0.0518 -0.5894 0.0519Capital-Size Ratio θ4 2.3054 2.1016 1.5037 1.0280

# Branches θ5 0.1561 0.0087 0.1577 0.0086Constant θ6 -6.6241 0.0280 -6.6283 0.0279

Table 2: Logit Demand Estimation

Demand elasticities w.r.t. size are also calculated. For small banks, the average elasticity

is 0.36 while for large banks, the average elasticity is 1.11. This large difference in elasticities

shows consumer’s preference for large network of branches. However, the average interest

rate elasticity is almost the same for small and large banks at 0.38 and 0.34 respectively.

These numbers are close to the price elasticities estimated by Dick (2008) for all banks using

MSA level data from 1993-1999 (0.30).

I perform three modifications to the base case as a robustness check. First, I estimate the

model with both demand and supply moments jointly but without any instrumental variables

for price and equity capital. Second, I estimate the demand parameters with demand-side

moments only and with instrumental variables. Third, I estimate the demand parameters

with demand-side moments only but without any instrumental variables.28 I find that using

instrumental variables changes the value of the parameter on the capital-size ratio variable to

some extent, and using the supply side moments is crucial for identification of the parameter

28Refer to the appendix 8.3 for the second and third case.

20

on price (θ1). This suggests that the correlation between unobserved quality and price is

weak. Overall, using supply side moments and instrumental variables doesn’t provide any

significant difference in the magnitudes of demand estimates.

Banks exercise market power by reducing their deposit interest rate. Since consumers

have a preference for a large network of branches, banks with large size exercise more market

power by lowering the deposit interest rate (hence reducing their interest expenses). Hence,

market power increases with size.

5.2 Cost Function Parameters

There are two stages of the cost function estimation. First, the parameter on equity

capital (βS4 and βL4 ) is estimated. Second, the parameters on the quadratic spline function

of network size (β1, β2 and β3) in the cost function are estimated using the moment inequal-

ity estimation. The moment inequality method would use all parameters estimated in the

previous stages.

Table 3 contains the estimates of β1, β2 and β3. The negative sign on β2 implies the

presence of cost efficiencies for small banks (less than 500 branches). For larger banks (more

than 500 branches), the parameter β3 is added into the cost function. The positive sign on

β3 implies that cost efficiencies are smaller in magnitude for larger banks because concavity

for larger banks is inferred by the sum: β2 + β3. As banks grow in size, cost efficiencies can

come from a reduction in risk management expenses and advertising expenditures because

these expenses don’t grow proportionally such as an expense for a advertisement on TV or

newspaper doesn’t grow with size.

The estimates imply that the physical capital and labor cost of setting up the first branch

is between 6.3780 million dollars and 6.6472 million dollars (β1 +β2). For the second branch

this cost drops by 9.8 thousand dollars to 12.8 thousand dollars (2β2). For each new branch,

up to the size of 500 branches this cost declines by a factor proportional to β2. The decline

in the marginal cost due to concavity is not big for banks in the very lower tail of the size

distribution, e.g. a bank with 10 branches will get a reduction of 2% in the marginal cost for

the next branch it adds. The cost savings become significant with somewhat larger banks,

e.g. a bank with 100 branches will get a reduction of 15% in the marginal cost for an extra

branch. Once I cross the barrier of 500 branches, concavity in the cost function is reduced.

At the lower end of β3, with a value of 0.0012, the large banks still have concavity in their

cost function. At the upper end of β3, with a value of 0.0121 the cost function for large

banks becomes convex with no cost efficiencies.

The inference results are also listed in table 3. The confidence intervals constructed using

21

the PPHI outer method and the moment selection method are similar qualitatively, while

the inner method of PPHI produces somewhat tighter confidence intervals. Although the

estimated set suggests the presence of cost efficiencies, some of the confidence intervals for

β2 and β3 contain zero implying that the evidence for cost efficiencies is not very strong in

the data. The main qualitative difference between this paper and earlier studies on cost

efficiencies in banking is that I have an explicit demand-side which allows me to account

for market power and consumers’ preference for size of the branch network. The existing

literature on banking scale economies has conflicting findings. Stiroh (2000) used 1991-1997

data to find that the largest bank holding companies have stronger cost efficiencies than the

smaller ones. Boyd and Graham (1998) examined the effects of mergers and found evidence of

cost efficiency gains for only the smallest banks. The gains disappeared quickly with increases

in size and were negative for larger banks. Hughes, Mester, and Moon (2001), Hughes, Lang,

Mester, and Moon (1996), Hughes and Mester (1998) find strong cost efficiencies for all banks

and the largest banks have slightly more cost efficiencies than rest of the banks. None of

the above mentioned papers controlled for market power at the local geographic level while

calculating cost efficiencies at the national level. These papers measure cost efficiencies as

the percentage change in profits/costs with unit change in size (measured by assets). It

is possible that the change in profits/costs with size could be due to the lowered interest

expenses on deposits (market power). This could be the reason why some of the above studies

find strong evidence of cost efficiencies. But once I control for market power, the evidence

of cost efficiencies becomes weak at best. Hence, the existing findings in the literature may

be misleading by ignoring the market power effect.29

Parameter Values 95 % Confidence IntervalLower Bound Upper Bound Inner C.I. Outer C.I. Moment Selection

β1 6.3844 6.6521 [4.4868 8.4056] [3.5815 8.6557] [6.2676 10.490]β2 -0.0064 -0.0049 [-0.0073 -0.0017] [-0.0103 0.0024] [-0.0066 0.0102]β3 0.0012 0.0121 [-0.0204 0.0196] [ -0.0182 0.0355 ] [-0.0137 0.0237]

Table 3: Cost Function Estimation(in million dollars)

The parameters on the equity capital are in the Table 4. This parameter measures the

interest rate on the funds generated from investors. Large banks pay an interest rate of

5.24% while the smaller banks have to pay a higher rate of 6.03%.30 These results support

29A similar argument can be constructed for ignoring consumers’ preference for size of a bank by the earlierpapers.

30Accounting for the error in demand parameters estimated in the first stage changes the parameters very

22

the fact that the larger banks are at an advantage when it comes to external funding. The

attractiveness of large banks to investors could be attributed to the fact that either their

portfolio is more diversified or their size signals safety.31 A similar result was found by Shull

and Hanweck (2001), where they find that the top 10 largest banks paid less for funds than

smaller banks.

Variable Parameter Value Std. ErrorEquity Capital(Small banks) βS4 0.0603 0.0250Equity Capital(Large banks) βL4 0.0524 0.0082

Table 4: Cost Function Estimation: Equity Capital Parameters

Using the parameters, I do some preliminary analysis to distinguish between diversi-

fication and size motives behind the difference in the cost of equity capital. I measure

diversification by the number of markets a bank is present in and I measure size by number

of branches. I run OLS regressions with the log of size and log of the number of markets

a bank is present in as the independent variables. Table 5 contains all the results. The

dependent variable is the log of the marginal cost in all three regressions which is calculated

from equation (2). Specification (1) contains log of the size as the only independent vari-

able. Specification (2) contains log of the number of markets a bank is present in as the

independent variable. Specification (3) has both of them as the independent variable. In

specification 3, which is the most general one, increase in size decreases the interest rate on

cost of funds. This shows some evidence of investors preferring to invest in larger banks over

the diversification incentive.

Parameter(Std. Error)Variable (1) (2) (3)log(Size) -0.0375(0.0038) - -0.0564(0.0059)

log(# Markets) - -0.04022(0.0087) 0.0559(0.0133)

Table 5: Cost Function Estimation: Parameters for variable marginal cost of equity capital.

Overall, the demand and cost estimates suggest that both market power at the geographic

marginally.31To answer this question, one has to incorporate risk into the bank’s profit function. By incorporating risk

I mean that bank’s care about both mean profits and the variance of profits so that portfolio diversificationcan be endogenized. I leave this topic for future research.

23

market level and a weak evidence of cost efficiencies at the firm level are present as banks

increase in size. To assess the relative importance of these two effects I simulate mergers

between two banks in the next section.

6 Industry Analysis and Counterfactual Experiments

6.1 Industry Analysis

The banking industry is one of the important industries in the U.S. with total assets of

approximately $11.7 trillion in 2006. For decades, commercial banks had been geographically

constrained by the McFadden Act of 1927 that prohibited them from operating across state

lines. Additionally, state laws often restricted banks ability to branch across county lines

and in many states prohibited branch banking entirely. The states deregulated their banking

laws at different times. Some deregulated as early as 1970 whereas others were deregulated

only when the Riegle-Neal Act was passed in 1994. The after effects of Riegle-Neal Act

on market structure are significant. This gradual deregulation has led to a consolidation of

banks over the last 30 years and is still ongoing. In 1990, there were 12,343 commercial banks

and 2,815 savings banks that were FDIC insured. In 2006, the number of commercial banks

was reduced to 7,402 while savings banks were reduced to 1,279. Since the financial crisis

started in 2007, the drop in the number of banks is also due to bank failures and the forced

mergers of failing institutions with healthy ones, as well as regular mergers between healthy

banks.32 In 2012, these numbers were reduced to 6,222 commercial banks and 1,024 savings

banks. Figure 2 shows the number of regular mergers (excluding corporate re-organization

mergers and failing bank mergers) from 2000-2010.33

Before the financial crisis in 2007, there were more than 100 regular mergers per year.

After 2007, the number of regular mergers drops but still there are more than 50 mergers

per year. The presence of so many mergers, suggests the need to study them in depth and

understand the driving forces behind these mergers and its effects on consumers.

32A regular merger is defined as a merger between two banks which are owned by separate bank holdingcompanies and neither of the banks is a failing institution.

33All bank mergers must be approved by one of the three federal bank regulators: Office of the Comptrollerof the Currency(OCC), Federal Deposit Insurance Corporation(FDIC) or Board of Governors of the FederalReserve System(FRB). Over the years OCC and FRB have published lesser details about the mergers makingit difficult to distinguish between regular mergers and corporate re-organization mergers. The histogram isbased on the numbers from FDIC alone.

24

Figure 2: Mergers between 2000 and 2010 in the US banking industry. Corporate re-organization mergers and failing bank mergers are excluded.

6.2 Merger Simulations

I use the estimated parameters to simulate some actual mergers between banks that

occurred in 2006 or later. For the set identified parameters, I use the mid-point of the set for

the following merger simulations.34 The objective of this exercise is to quantify the effects

of market power, cost efficiencies and equity capital on the profitability of a merger. I also

calculate the change in consumer welfare due to the decrease in deposit rates and increase

in network size because of the merger.

In the following experiments, revenues from loans are split in two categories: loans funded

through deposits (∑

mDmsjm(θ)(P lj − P d

jm)) and loans funded through equity capital (P ljkj).

Costs are also split in two categories: operating costs (β1nj +β2n2j +β3I(nj > X)(nj −X)2)

and cost of raising equity capital (βS4 kj or βL4 kj). The numbers in the tables below are

the difference between the merged entity with joint profit maximization and consolidated

numbers of the two banks with pre-merger values.

There are two demand-side effects due to a merger. First, consumers get a better quality

product as they prefer larger network of branches (demand synergies). Second, consumers

get a lower interest rate on deposits as firms re-optimize deposit rates. I attribute this second

effect to market power. To isolate market power, I have to separate these two effects so that

I can measure the effect of price alone. I explain my calculation of market power using a

simple example. Say, bank 1 and bank 2 merge into a bank 12. Each bank’s network and

34I am currently working on robustness to this choice of mid-point of the parameter set. I plan to uniformlysample the parameter set and run the counterfactual experiments for each parameter.

25

deposit rate are denoted by ni and pi respectively, where i = 1, 2 or 12. The merged bank’s

network, n12, is the combined network of bank 1 and bank 2, while the deposit rate, p12,

is obtained by re-optimizing prices with the combined network of branches. Let the loan

revenues funded through deposits for bank i be denoted by r(pi, ni)35. I can measure the

combined effect of demand synergies and market power by

E1 ≡ r(p12, n12)− r(p1, n1)− r(p2, n2). (7)

E1 measures the combined effect because consumer’s utility depends on both prices and

total size of the network. To measure the demand synergies component of the total effect, I

assume a hypothetical scenario. I assume that both the merging banks, 1 and 2, are present

in the economy with each bank having all characteristics of the merged bank except the

deposit rate. In this setup, both the merging banks re-optimize prices.36 Let this price be

denoted by p. This setup will measure the effect of the network on profits without any price

effects coming from the reduction of competition. Note that in this hypothetical scenario

there is no reduction in the number of players in a market. I quantify demand synergies by

measuring

E2 ≡ r(p, n12)− r(p1, n1)− r(p2, n2), (8)

where r(p, n12) is the revenue of one of the merging banks with the combined network.

Finally, to calculate the market power effect, I subtract the equation (8) from the equation

(7) (E1 − E2). Using this approach, there is no market power effect in the markets where

the merging banks do not overlap. This is inline with the fact that market power is a local

geographic phenomenon. A similar calculation is done for consumer surplus to isolate the

market power from the combined effect.

The cost efficiency in a merger between two banks with n1 and n2 branches is quantified

by,

CE(n1, n2) ≡ [β1(n1 + n2) + β2(n1 + n2)2 + β3I((n1 + n2) > X)((n1 + n2)−X)2]

−[β1n1 + β2n21 + β3I(n1 > X)(n1 −X)2]− [β1n2 + β2n

22 + β3I(n2 > X)(n2 −X)2]. (9)

The first term in CE(n1, n2) is the operating cost of the merged entity, the second and

35In the real calculation there are other variables also, but to simplify the exposition I omit them in thisexample.

36Note that both the merging banks will choose the same price.

26

third terms are operating costs of the two merging banks. The concavity in the quadratic

function generates cost savings for the merged bank. Similarly, cost savings from equity

capital (say for a merger between two large banks) are measured by,

E3 ≡ βL4 k12 − βL4 k1 − βL4 k2, (10)

where k12 is the equity capital corresponding to the merged bank. It is important to note

that the choice of mid-point for the cost parameters will only affect the measurement of cost

savings, the quantification of market power and demand synergies is immune to this choice.

While simulating the mergers, I need to make a choice for the loan rate and the un-

observed bank-market quality (ξjm) of the merged entity. I choose the maximum loan-rate

and maximum ξjm of the two banks for the merged entity.37 Using the maximum value

for unobserved quality is roughly equivalent to using the unobserved quality of the larger

bank. Since the larger bank is usually the acquiring bank, it is reasonable to assume that

the brand-market fixed effects of the merged entity are that of the acquiring bank.

I simulate two mergers in each of the following three categories: small bank and small

bank, small bank and large bank, large bank and large bank. From 2006 onwards, there

have been more than 50 regular mergers every year. In many of these mergers, the acquired

bank had less than 5 branches. For simulating the mergers involving small banks (less

than 500 branches), I chose the ones where the bank size was not too small (more than

30 branches) so that the change in magnitudes for important variables is significant. For

simulating mergers between two large banks(more than 500 branches), I pick one regular

merger (Regions Bank and Amsouth Bank) and one failing bank merger (Wells Fargo and

Wachovia). When Wachovia bank was collapsing in the financial crisis, it was forced by

FDIC to sell itself.

There are some common elements in each merger simulation. The network of branches of

the two banks are merged exogenously and after that the merged entity chooses new equity

capital and deposit interest rates.38 For the merged bank, the deposit rate in each market

decreases and market share increases compared to the individual banks. This drives up

the gains from revenue sources in a merger. Equity capital of the merged bank is strongly

correlated with size, hence the merged bank has a higher equity capital than each of the

merging banks.

37Results are very stable even if I pick minimum loan rate.38Note that other banks are not allowed to re-optimize their network choice in response to the merger.

This is not feasible in the current setup as I have not solved for the bank’s policy function.

27

6.3 Mergers between two small banks

The first merger simulation in this category is between Prosperity Bank,TX (75 branches

in 6 markets) and State Bank,TX (37 branches in 5 markets) which occurred in 2007. The

two banks overlap in 4 markets. Prosperity bank is headquartered in El Campto, TX and

State Bank was headquartered in La Grange, TX. Due to the merger, total revenues increased

by 30.1 million dollars, most of which is contributed by equity capital. The merged bank

generates extra 8.8 million dollars in revenue from deposits, of which only 1.45 million is

attributed to market power. This shows that there are strong demand synergies (preference

for large networks) compared to the market power effect. There are significant cost efficiencies

leading to a savings of 20.8 million dollars and most of it comes from reduction in operating

expenses. Cost efficiencies seem to be playing a more important role in this merger compared

to market power. Overall, consumer surplus decreases by a small amount indicating that

that the utility from a better quality product (bigger size of the merged entity) is not offset

by the decreased deposit rates due to market power. The loss in consumer surplus due to

market power is -0.22 basis points. In this setup, consumer surplus is the utility, in interest

rate terms, that the consumer receives.

$ Change(million dollars) % ChangeChange in Revenues

From deposits(Combined effect) 8.88 4.60%From deposits(Market Power effect) 1.45

From Equity 21.20 11.10%

Change in Total Revenues 30.08 7.90%Change in Cost

From Operating Cost -31.60 -4.60%From Equity 10.80 6.10%

Change in Total Costs -20.80 -2.40%Change in Total Profits 50.90 10.5%

Change in Total Consumer Surplus -0.098 -0.09%CS loss due to Market Power -0.22

Table 6: Merger Simulation : Prosperity Bank,TX (75 branches in 6 markets) and StateBank,TX (37 branches 5 markets).

The second merger in this category is between First Tennessee Bank (232 branches in

24 markets) and Sterling Bank (41 branches in 3 markets) that occurred in 2006. First

Tennessee Bank is headquartered in Memphis, TN and Sterling Bank was headquartered in

Houston, TX. There is only 1 overlapping market among the merging banks. The merged

entity is able to generate an extra 52.9 million dollars through offering a better quality

28

product and charging lower deposit rates. Since there is only one overlapping market out of

the total 26 markets the merged firm is present in, there is almost no market power effect

here. There is extra revenue generated through equity capital as well (39.6 million dollars)

but that is almost offset by the extra cost (36.3 million dollars) incurred to raise that much

equity. Operating costs drop by 108 million dollars due to cost efficiencies resulting from the

merger. In this simulation, cost efficiencies seem to be an important driver of the merger.

Overall, consumers benefit from this merger as the depositors of the smaller bank get an big

increase in the quality of the product because of the added network of a bigger bank. The

loss in consumer surplus due to market power is -0.03 basis points. Note that this magnitude

is much smaller than the first merger in this category. This happens because less overlap in

markets reduces the market power effect.

$ Change(million dollars) % ChangeChange in Revenues

From deposits(Combined effect) 52.90 5.3%From deposits(Market Power effect) 0.0022

From Equity 39.60 8.3%

Change in Total Revenues 92.50 6.3%Change in Cost

From Operating Cost -108 -7.4%From Equity 36.30 7.2%

Change in Total Costs -71.70 -3.7%Change in Total Profits 164 32.7%

Change in Total Consumer Surplus 0.094 0.03%CS loss due to Market Power -0.03

Table 7: Merger Simulation : First Tennessee Bank (232 branches in 24 markets) and SterlingBank (41 branches in 3 markets) .

Overall, in mergers between two small banks cost efficiencies play a more important role.

The consumer welfare either drops or increases by a small amount. This happens because

the banks in these mergers are small and consumers don’t benefit too much from the demand

synergies. Note that since prices are chosen in the last stage of the game, cost savings are

never passed on to the consumers.

6.4 Mergers between a small bank and a large bank

The first merger in this category is between PNC bank (837 branches in 23 markets)

and Mercantile-Safe Deposit and Trust Company (195 branches in 7 markets) that occurred

in 2007. PNC bank is headquartered in Pittsburgh, PA and Mercantile-Safe Deposit and

29

Trust Company was headquartered in Baltimore, MD. These banks overlap in 3 markets. As

a result of the merger, revenues from deposits increase by 54.8% which translates into 835

million dollars. Out of the 835 million dollars, only 27.4 million dollars can be attributed

to market power. This implies presence of strong demand synergies. Revenue from equity

comes around to be 500 million dollars which is largely offset by the cost of raising equity,

440 million dollars. Cost savings due to merger synergies are 22.6 % which in dollar terms

are 738 million dollars. Total revenues increase by 1.3 billion dollars (33.2 %) whereas total

costs decrease by 298 million dollars (5.1%) only. In this simulation, I can clearly see cost

efficiencies dominating market power as a possible driving force for merger. This happens

because there are very few overlapping markets. Consumer surplus increases by 6.67 basis

points indicating that the effect of increased market power is dominated by a better quality

product (larger bank). The loss in consumer surplus due to market power is -0.31 basis

points. Note that the loss in consumer surplus here is larger than both the small-small

mergers, but is still offset by demand synergies.

$ Change(million dollars) % ChangeChange in Revenues

From deposits(Combined effect) 835 54.8%From deposits(Market Power effect) 27.4

From Equity 500 20.1%

Change in Total Revenues 1,335 33.2%Change in Cost

From Operating Cost -738 -22.6%From Equity 440 17.3%

Change in Total Costs -298 -5.1%Change in Total Profits 1,633 90.6%

Change in Consumer Surplus 6.67 1.92 %CS loss due to Market Power -0.31

Table 8: Merger Simulation : PNC Bank (837 branches in 23 markets) and Mercantile-SafeDeposit and Trust Company (195 branches in 7 markets).

The second merger in this category is between Wells Fargo Bank (2,613 branches in 127

markets) and Greater Bay Bank (41 branches in 5 markets) that occurred in 2008. Wells

Fargo Bank is headquartered in San Francisco, CA and Greater Bay Bank was headquartered

in Palo Alto, CA. Before the merger, Wells Fargo was present in all the 5 markets where

Greater Bay Bank was located. The revenue from loans funded by deposits increases by

1.01 billion dollars (11.6 %). Out of these 1.01 billion dollars, 339 million dollars can be

attributed to market power. The revenues from loans funded by equity is almost balanced

30

by the cost incurred to raise that equity. Operating costs decline by 62 million dollars (-0.8

%) which is really small as compared to the gains from market power. Overall the total

revenues go up by 1.26 billion dollars while total costs also rise by 170.80 million dollars.

Hence, the profitability of this merger is driven by market power more. The large market

power effects are present due to a lot of overlapping markets between the two merging banks.

The total consumer surplus goes up suggesting that the effect of decreased interest rates is

more than offset by a better quality product (larger bank). This happens because the utility

of consumers of Greater Bay Bank increases a lot as the network size increases from 41 to

2,654 due to the merger. The loss in consumer surplus due to market power alone is -1.3

basis points. Note that this consumer surplus loss is larger than the loss in both small-small

bank mergers.

Overall, the loss in consumer surplus due to market power is more compared to the small-

small bank mergers. But the change in total consumer surplus is larger compared to the

small-small bank mergers. Hence, I can conclude that although market power increases when

large banks are merging, demand synergies increase at a greater rate. The cost efficiencies

seems to be driving the mergers when there is not a lot of overlap between merging banks.

When a large bank is involved in a merger, banks are practically breaking even in terms of

equity capital. This happens because of two reasons. First, equity capital is costlier than

deposits so banks with large network of branches don’t want to raise more equity to fund

loans. Second, large banks have strong brand effects and don’t need to signal safety to

depositors through capital reserves.

$ Change(million dollars) % ChangeChange in Revenues

From deposits(Combined effect) 1,010 11.6%From deposits(Market Power effect) 339

From Equity 247 1.9%

Change in Total Revenues 1,257 5.8%Change in Cost

From Operating Cost -62.20 -0.8%From Equity 233 1.9%

Change in Total Costs 170.8 0.8%Change in Total Profits 1,086 96.9%

Change in Total Consumer Surplus 16.13 1.04%CS loss due to Market Power -1.30

Table 9: Merger Simulation : Wells Fargo Bank (2,613 branches in 127 markets) and GreaterBay Bank (41 branches in 5 markets).

31

6.5 Mergers between two large banks

The first merger in this category is between Regions Bank (971 branches in 89 markets)

and Amsouth Bank (561 branches in 44 markets) that occurred in 2006. They overlap in

36 markets. Both Regions Bank and Amsouth Bank are headquartered in Birmingham, AL.

The revenue from loans funded by deposits increased by 3.8 billion dollars (158%) while

operating costs decreased only by 648 million dollars (15%). Out of the 3.8 billion dollars

increase in revenue, 1.26 billion dollars can be attributed to market power. This is largely

driven by a big overlap of markets. Like the previous merger simulations, the change in

revenue and cost from equity capital roughly balance each other. Overall, total revenues

increase by 5 billion dollars while total costs increase by 853 million dollars. At this point,

the concavity in the cost function has been diminished by a significant amount. Clearly, the

major driver behind this merger is market power. The total consumer surplus increases by

4.25 basis points suggesting that the effect of decreased interest rates is more than offset by

a better quality product (larger bank). The loss in consumer surplus due to market power

is -13.2 basis points which is larger compared to any of the mergers in last two categories.

$ Change(million dollars) % ChangeChange in Revenues

From deposits(Combined effect) 3,794 158.0%From deposits(Market Power effect) 1,260

From Equity 1,451 44.7%

Change in Total Revenues 5,054 92.8%Change in Cost

From Operating Cost -648 -15.0%From Equity 1,501 37.9%

Change in Total Costs 853 10.3%Change in Total Profits 4,201 92.8%

Change in Total Consumer Surplus 4.25 0.34%CS loss due to Market Power -13.20

Table 10: Merger Simulation : Regions Bank (971 branches in 89 markets) and AmsouthBank (561 branches in 44 markets).

The second merger in this category is between Wells Fargo Bank (2,613 branches in 127

markets) and Wachovia bank (2,795 branches 105 markets) which occurred in 2008. The

merging banks overlap in only 8 markets. Wachovia bank was headquartered in Charlotte,

North Carolina and Wells Fargo Bank is headquartered in San Francisco, CA. This merger

took place during the financial crisis and it was a FDIC forced merger. The difference in

32

$ Change(million dollars) % ChangeChange in Revenues

From deposits(Combined effect) 4,130 24.9%From deposits(Market Power effect) 222

From Equity 6,828 24.5%

Change in Total Revenues 10,958 24.7%Change in Cost

From Operating Cost 11,856 72.8%From Equity 7,415 28.3%

Change in Total Costs 19,271 45.4%Change in Total Profits -8,313 -24.7%

Change in Total Consumer Surplus 22.41 0.82%CS loss due to market power -0.80

Table 11: Merger Simulation : Wells Fargo Bank(2,613 branches) and Wachovia bank(2,795branches).

total revenues from this merger is 10.9 billion dollars while an extra total cost of 19.2 billion

dollars needs to be incurred. This shows that the merger was not profitable under normal

scenarios. Total consumer surplus went up by 22.4 basis points. This particular simulation

exercise also acts as a robustness check for our parameters.

The driving force behind merger between two large banks is mostly market power. Also,

the reason why cost efficiencies are not large in this category of mergers is because the

parameter β3 is important and decreases the concavity in the cost function. Also, consumer

welfare goes up in the merger between two large banks due to consumer’s preference for a

larger network of branches.

7 Conclusion

This paper quantifies the degree of market power and cost efficiencies for U.S. banks.

This paper develops a model of consumer behavior and firm choice where market power is

a geographically local phenomenon whereas cost efficiencies are realized at the firm level. I

develop a three-stage empirical model in which consumers choose banks for deposit services

and banks choose the network of branches, equity capital and deposit rates. To estimate the

cost parameters related to the network choice, I use moment inequality methods.

Demand estimates suggest that consumers prefer large, more capitalized banks and that

market power increases with size. After controlling for market power, the evidence for cost

efficiencies is weak for smaller banks (less than 500 branches), and it declines as banks get

larger (more than 500 branches). This is a contribution to the existing literature which

33

doesn’t control for market power and size effects in calculating cost efficiencies in the US

banking industry. I also find that smaller banks are at an disadvantage when it comes to

borrowing money from external sources (raising equity capital).

Using the estimated parameters, I simulate mergers in three categories: between two small

banks, a small bank and a large bank and between two large banks. For mergers between

two small banks, cost efficiencies play an important role. For a merger between a small and a

large bank, the extra revenue generated by market power is larger than the cost savings when

there is a lot of overlap in the markets of merging banks. And for merger between two large

banks, market power effect dominates the cost efficiencies effect. Consumer surplus always

goes up in the mergers involving large banks. This happens because the market power effect

is dominated by the consumer’s preference for large network of branches. Hence, just looking

at the market power or cost efficiency is not sufficient for approving/declining a merger. The

fact that consumers have a better product at disposal should be taken into account.

8 Appendix

8.1 Sufficiency conditions for PPHI estimator

To use the PPHI estimator, the weighting function and errors should satisfy two suffi-

ciency conditions.

Condition 1: E[∑

j

∑n′jh(n

′j;nj, Ij)νj,nj ,n

′j] ≥ 0

Condition 2: E[∑

j

∑n′jh(n

′j;nj, Ij)∆γj,nj ,n

′j] ≤ 0

Using the information that E[νj,nj ,n′j|Ij] = 0 and since Ij doesn’t contain any information

about rivals condition 1 is trivially satisfied.

Condition 2 is also trivially satisfied as ∆γj,nj ,n′j

equals zero. In my model, the structural

error γj is fixed before the network choice.

8.2 Inference using Generalized Moment Selection

Following steps are used in sequence to calculate the confidence sets,

1. Form a 3-dimensional grid of points in (β1, β2, β3) space which extends well beyond the

identified set.39

39The grid is constructed so that the null hypothesis is rejected at the end points of the grid. In otherwords, the end points of the grid are such that they are not in the confidence interval.

34

2. At each grid point βg, evaluate the objective function: Q(βg) = ||(D̂M−1/2

s(βg))−||,where D̂M is a diagonal matrix with variance of moments on its diagonal.

3. At each grid point a critical value is calculated through simulation, cα(βg).

4. If Q(βg) ≤ cα(βg), then βg is in confidence set.

The calculation of critical value is the most important step in the above method. To

compute the critical value, an approximation to the distribution of objective function under

the null is simulated. The distribution of the sample moments in the expression of objective

function are simulated by a normal with mean zero and variance calculated using the data

across markets. Next, of all the simulated moments under the null, only the nearly binding

moments at each βg enter the expression for simulating the objective function. A moment

k is defined to be nearly binding if√ME(sk(βg))/σ̂(sk(βg)) <

√2ln(ln(M)) , where the

expectation operator is replaced by its sample analog and M is the total number of markets.

8.3 Robustness for demand estimates

The demand estimates in the paper are calculated using both demand and supply side

moments. The table below has estimates with only demand side moments. This acts as a

robustness check for the baseline results used in the paper.

Demand moments only Demand moments only(with IV) (without IV)

Variables Parameter Value Std. Error Value Std. ErrorPrice θ1 42.2818 25.5104 38.2860 5.2018

Branch Density θ2 22.3717 0.5920 22.2994 0.58691-Branch Dummy θ3 -0.5763 0.0521 -0.5763 0.0521Capital-Size Ratio θ4 1.5304 2.3211 1.4429 1.0357

# Branches θ5 0.1605 0.0106 0.1604 0.0087Constant θ6 -6.9207 0.4182 -6.8580 0.0888

Table 12: Logit Demand Estimation : With demand moments only

35

8.4 First order condition of equity capital

Here is the simplifying algebra of the first order condition of the equity capital (kj).

Assume that j is a small bank.

∂Πj

∂kj= 0

∑m

Dm

θ1[

∂sjm∂kj

1− sjm+

sjm∂sjm∂kj

(1− sjm)2] + P l

j −∂C(nj, kj)

∂kj= 0

∑m

Dm

θ1[

∂sjm∂kj

1− sjm+

sjm∂sjm∂kj

(1− sjm)2] + P l

j − βS4 = 0

Hj + P lj − βS4 = 0

where Hj =∑

m

Dm

θ1[

∂sjm∂kj

1− sjm+

sjm∂sjm∂kj

(1− sjm)2] + P l

j .

Using the logit error assumption in the utility of the consumer I can simplify∂sjm∂kj

,

∂sjm∂kj

=∂sjm

∂(kj/nj)

∂(kj/nj)

∂kj=θ4sjm(1− sjm)

nj.

Note that nj is treated as constant in the above equation, because in the second stage when

the banks are choosing kj, bank size (nj) has already been decided in the first stage. Hence

I have,

Hj =∑m

Dm

θ1[

∂sjm∂kj

1− sjm+

sjm∂sjm∂kj

(1− sjm)2] + P l

j

Hj =∑m

Dm

θ1[θ4sjm(1− sjm)

nj(1− sjm)+θ4s

2jm(1− sjm)

nj(1− sjm)2] + P l

j

Hj =∑m

Dm

θ1[θ4sjm(1− sjm)

nj(1− sjm)+θ4s

2jm(1− sjm)

nj(1− sjm)2] + P l

j

Hj =∑m

Dm

θ1[

θ4sjmnj(1− sjm)

] + P lj

Substituting the value of Hj in the FOC I get,

36

∑m

Dm

θ1[

θ4sjmnj(1− sjm)

] + P lj = βS4 .

The above equation forms the basis for estimation of βS4 . A similar calculation can be

done for large banks as well.

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